Z94.1 - Analytical Techniques & Operations Research Terminology

PARAMETER. This word occurs in its customary mathematical meaning of an unknown quantity which may vary over a certain set of values. In statistics it most usually occurs in expressions defining frequency distributions (population parameters) or in models describing a stochastic situation (e.g., regression parameters). The domain of permissible variation of the parameter defines the class of population or model under consideration. [22]

PARAMETRIC PROGRAMMING. A method for investigating the effect on an optimal linear programming solution of a sequence of proportionate changes in the elements of a single row or column of the matrix. Most commonly the method is applied to either the objective function row or the right-hand-side column. [19]

PARETO OPTIMALITY (NON-DOMINATED SOLUTION). A solution to a multicriterion problem for which there exists no other feasible solution which has an equal or superior criterion value for each criterion.

PARTITION. To separate a linear program into related subprograms. More generally to divide a set into nonintersecting subsets. [19]

PAYOFF. The amount received by one of the players in a play of a game. [21]

PAYOFF FUNCTION. For a two-person zero-sum game, the payoff function M is the function for which M (x, y) (positive or negative) is the amount paid by the minimizing player to the maximizing player in case the maximizing player uses pure strategy x and minimizing player uses pure strategy y. [21].

PAYOFF MATRIX. A two dimensional array {Pij} in which the columns (or rows) represent the available strategies, the rows (or columns) represent the states of nature and the entries Pij represent the outcomes (on some scale of measure) for the ith column and the jth row.

PERMANENTLY FEASIBLE SET (STOCHASTIC PROGRAMMING). Consider a linear program with random parameters. Here one considers only the convex set of those vectors X which will be feasible regardless of the subsequently observed parameters of the linear program. For each (A,b) the X’s satisfying AX = b, X ≥ 0 form a convex polyhedron. The set of permanently feasible X’s are those that are elements of the intersection of these polyhedra where the intersection is overall A,b. [19]

PERSONNEL-ASSIGNMENT PROBLEM. (See ASSIGNMENT PROBLEM.) [15]

PERSONNEL SUBSYSTEM. A management concept which considers that functional part of a system which provides, through effective development and implementation of its various elements, the specified human performance necessary in the operation, maintenance, support, and control of the system in a specified environment.

PERTURBATION TECHNIQUES. The process of modifying slightly the right-hand-side coefficients of a linear programming problem to insure against the possibility of cycling. [15]

P-HARD PROBLEMS. Problem for which algorithms have been found that can be solved in a time proportional to a polynomial function of the value of a measure of the problem size or complexity.

PHASE I. The mathematical procedure used by the simplex algorithm, two-phase method, to find a first feasible solution to a linear programming problem. [19]

PHASE II. The mathematical procedure used by the simplex algorithm, two-phase method, to find an optimal basic feasible solution, given a first basic feasible solution. [19]

PHASE PROCESS. A renewal process where each independent and identically distributed interval-length distribution is the distribution of a phase random variable.

PHASE QUEUING MODEL. A queueing model in which either the arrival or service process or both is a phase process. This class of queueing model is a generalization of the so-called "method of stages," where the service-time or arrival-time distributions were either hyperexponential, Erland, or Coxian. Computational results for phase queueing models allow for numerical evaluation of complex models.

PHASE-TYPE RANDOM VARIABLE. The time-til absorption in a finite-state Markov Process having exactly one absorption state. Phase-type random variables are Markovian, thus their use in probability modeling allows for relatively easy computation in the form of small matrix inversions or numerical integration. Phase distributions are also dense in the set of distributions having support on the non-negative real-number line; meaning that phase distributions can take on any arbitrary shape. Their flexibility in shape and computational convenience has led to their adoption in a wide variety of probability models.

PIECEWISE LINEAR APPROXIMATION. The division of the domain of definition of a function into subregions, and the replacement of the function by some closefitting linear function in each subregion. [19]

PIVOT COLUMN. The column of the matrix containing the pivot element. In a linear programming iteration, it is the column associated with the entering variable (nonbasic variable picked to become basic). [19]

PIVOT ELEMENT. The element in a matrix found at the intersection of the pivot column and pivot row.

PIVOT ROW. The row of the matrix containing the pivot element. In a linear programming iteration, it is the row associated with the departing variable (basic variable picked to become nonbasic). [19]

PIVOT STEP. A step consisting of a single transformation of the matrix in a pivotal method of reduction of a set of linear equations. [19]

PIVOTAL METHOD. One of the methods used in the solution of sets of linear equations, in which some particular coefficient plays a dominant role at each stage of the elimination process. One such method, which forms the base of the ordinary simplex method of linear programming, is called the method of multiplication and subtraction. A step in this method involves transforming the equations so that a designated variable Xs shall appear only in the rth equation and no others. To accomplish this, row r of the matrix is first divided by ars and suitable multiples of this row are subtracted from all other rows so that the coefficient of xs in the other equations is zero. The coefficient ars appears in the computation of each new coefficient, it is called the pivot element. [12]

POLAR CONE. A cone in which all vectors make an angle of 90° or less with the generating vector.

POSITIVE DEFINITE (OR SEMIDEFINITE) QUADRATIC FORM. A quadratic form is called positive definite if X TAX > 0 for every X except X = 0. It is called positive semidefinite if X TAX ≥ 0 for every X and there exist X ≠ 0 for which XT AX = 0. [17]

PRECISION. The closeness of agreement between randomly selected individual measurements or test results.

PREDICTED. Expected at some future date, on the basis of analysis of past experience. [28]

PREEMPTIVE PRIORITY (PREEMPTIVE SERVICE). A queue discipline in which the arrival of a higher priority while a lower priority is in service requires the return to the queue of the lower priority. There are two cases: a.) Repeat rule: The ejected item returns to service, having lost all service before ejection; b.) Preemptive resume rule: The ejected item resumes at the point of service where it was interrupted. [33]

PROBABILITY. A basic concept which may be taken either as expressing in some way a “degree of belief,” or as the limiting frequency in an infinite random series. Both approaches lead to much the same calculus of probabilities. [22:226, 6:220, 25:18, 31:59]

PROCESS. Any set of conditions or causes which work together to produce a given result. [32]

PRODUCER'S RISK. For a given sampling plan, the probability of not accepting a lot the quality of which has a designated numerical value representing a level which it is generally desired to accept. Usually the designated value will be the Acceptable Quality Level (AQL).

PRODUCT FORM OF INVERSE (COMPUTING FORM). A computationally efficient (for sparse matrices) way of updating only the required portion of the inverse; often used in revised simplex (q.v.) codes due to its calculation and input-output characteristics. [19]

PROFIT RANGE. The interval (FORMULA MISSING) where c is a profit coefficient in the objective function, and the two endpoints are points where a change of basis must fist occur to maintain optimality.

PROGRAMMING. The investigation of the structure and state of a system and the objective to be fulfilled in order to construct a statement of the actions to be performed, their timing, and their quantity (called a program or schedule) which will permit the system to move from a given status toward the defined objective. [11]

PROGRAMMING PROBLEMS. Programming problems are concerned with the efficient use or allocation of limited resources to meet desired objectives. [15]

PURE STRATEGY. A mixed strategy (q.v.) which has all frequencies equal to zero except one which equals unity. [11]